Update on Overleaf.

1 files changed, 28 insertions, 25 deletions

diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex
index d8f7f87..4932050 100644
--- a/frsb_kac_new.tex
+++ b/frsb_kac_new.tex
@@ -10,27 +10,24 @@ or Full solution for the counting of saddles of mean-field glass models}
 \maketitle
 \begin{abstract}
     We derive the general solution for the computation of saddle points
-    of  mean-field complex landscapes. The solution incorporates Parisi's solution
-    for the ground state, as it should.
+    of  mean-field complex landscapes. The solution incorporates Parisi's solution for the ground state, as it should.
 \end{abstract}
 \section{Introduction}
 
 The computation of the number of metastable states of mean field spin glasses
 goes back to the beginning of the field. Over forty years ago,
 Bray and Moore \cite{Bray_1980_Metastable} attempted the first calculation for 
- the Sherrington--Kirkpatrick model, a paper remarkable for being the first practical application of a replica symmetry breaking scheme. As became clear when the actual
- ground-state of the model was computed by Parisi \cite{Parisi_1979_Infinite}, the Bray--Moore result
- was not exact, and
-in fact  the problem has been open 
-ever since.
-Indeed, to this date the program of computing the number of saddles of a mean-field 
+ the Sherrington--Kirkpatrick model, in a paper remarkable for being the first practical application of a replica symmetry breaking scheme. As became clear when the actual ground-state of the model was computed by Parisi \cite{Parisi_1979_Infinite} with a different scheme, the Bray--Moore result
+ was not exact, and in fact  the problem has been open ever since.
+To this date the program of computing the number of saddles of a mean-field 
 glass has been only  carried out for a small subset of models.
-These include most notably the $p$-spin model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer}. 
+These include most notably the (pure) $p$-spin model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer}. 
 The problem of studying the critical points of these landscapes
 has evolved into an active field in probability theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity, BenArous_2019_Geometry}
 
 In this paper we present what we argue is the general replica ansatz for the 
-computation of the number of saddles of generic mean-field models,  including the Sherrington--Kirkpatrick model. It incorporates  the Parisi solution as the limit of lowest states, as it should.
+computation of the number of saddles of generic mean-field models,  including the Sherrington--Kirkpatrick model. It reproduces  the Parisi result in the limit
+of small temperature for the lowest states, as it should.
 
 
 \section{The model}
@@ -442,30 +439,36 @@ which is precisely \eqref{eq:ground.state.free.energy} with $R_d=z$ and $\hat\ep
 
 {\em We arrive at one of the main results of our paper: a $(k-1)$-RSB ansatz in Kac--Rice will predict the correct ground state energy for a model whose equilibrium state at small temperatures is $k$-RSB }
 
-\section{Ultrametricity rediscovered}
 
-(not sure)
+\section{Ultrametricity rediscovered}
 
+TENTATIVE BUT INTERESTING
 
-Three states chosen at the same energy share some common information if there is some `frozen' element common to all. Suppose we choose randomly 
-these states but restrict to those whose overlaps
-take values $Q_{12}$ and $Q_{13}$. Unlike an equilibrium situation, where the Gibbs measure allows us to find such pairs (in a FRSB case) the cost in probability of this in the present case will be exponential.
-Once conditioned this way, we compute $Q_{23}= \min(Q_{12},Q_{13})$
+The frozen phase for a given index ${\cal{I}}$ is the one for values of $\hat \beta> \hat \beta_{freeze}^{\cal{I}}$. 
 
+[Jaron: does $\hat \beta^I_{freeze}$ have a relation to the largest $x$  of the ansatz? If so, it would give an interesting interpretation for everything]
+ 
+ The complexity of that index is zero, and we are looking at the lowest saddles 
+in the problem, a question that to the best of our knowledge has not been discussed
+in the Kac-Rice context -- for good reason, since the complexity - the original motivation - is zero.
+However, our ansatz tells us something of the actual organization of  the lowest saddles of each index in phase space. 
 
 
 
 \section{Conclusion}
-We have constructed a  replica solution for the general Kac-Rice problem, including systems
+We have constructed a  replica solution for the general  problem of finding saddles of random mean-field landscapes, including systems
 with many steps of RSB. 
-The main results of this paper are the ansatz \ref{ansatz} and the check that the lowest energy
-is the correct one obtained with the usual Parisi ansatz. 
-For systems with full RSB, we find that minima are, at all energy densities above the ground state one, exponentially subdominant with respect to saddles.
-It remains to exploit the construction to study general landscapes in more detail. 
+The main results of this paper are the ansatz (\ref{ansatz}) and the check that the lowest energy is the correct one obtained with the usual Parisi ansatz. 
+For systems with full RSB, we find that minima are, at all energy densities above the ground state, exponentially subdominant with respect to saddles.
+The solution contains valuable geometric information that has yet to be
+extracted in all detail.
 
+\paragraph{Funding information}
+J K-D and J K are supported by the Simons Foundation Grant No. 454943.
 
+\begin{appendix}
 
-\section{Appendix: RSB for the Gibbs-Boltzmann measure}
+\section{RSB for the Gibbs-Boltzmann measure}
 
 \begin{equation}
   \beta F=-\frac12\lim_{n\to0}\frac1n\left(\beta^2\sum_{ab}f(Q_{ab})+\log\det Q\right)-\frac12\log S_\infty
@@ -575,7 +578,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
 
 
 
-\section{Appendix: RSB for the Kac-Rice integral}
+\section{ RSB for the Kac-Rice integral}
 
 
 \subsection{Solution}
@@ -721,7 +724,7 @@ complexity in the ground state are
   D_d=R_d\hat\epsilon
 \end{align}
 
-\section{Appendix: a motivation for the ansatz}
+\section{ A motivation for the ansatz}
 
  We may encode the original variables in a superspace variable:
 \begin{equation}
@@ -776,7 +779,7 @@ Not surprisingly, and for the same reason as in the quantum case, this ansatz cl
  -i\bar\theta_1\theta_1\bar\theta_2\theta_2D_1R_2
  \end{aligned}
 \end{equation}
-
+\end{appendix}
 
 
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